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Activity 2.8.4 .
Evaluate each of the following limits. If you use L’Hôpital’s Rule, indicate where it was used, and be certain its hypotheses are met before you apply it.
(a)
\(\displaystyle \lim_{x \to \infty} \frac{x}{\ln(x)}\)
(b)
\(\displaystyle \lim_{x \to \infty} \frac{e^{x} + x}{2e^{x} + x^2}\)
(c)
\(\displaystyle \lim_{x \to 0^+} \frac{\ln(x)}{\frac{1}{x}}\)
(d)
\(\displaystyle \lim_{x \to \frac{\pi}{2}^-} \frac{\tan(x)}{x-\frac{\pi}{2}}\)
(e)
\(\displaystyle \lim_{x \to \infty} xe^{-x}\)
